def setup_RPn_cost(D, return_grad=False):
"""Create the cost functions for pymanopt.
The cost function is given by
F(X) = (1/2)*||W * |X^T X| - cos(D))||^2
Where `W` is the weight matrix accounting for removing the arccos term.
Currently only returns the cost. For better performance, could add gradient
as a return value.
Parameters
----------
D : ndarray (n, n)
Matrix of target distances.
Returns
-------
cost : function
Weighted Frobenius norm cost function.
egrad : function
Gradient (in Euclidean space) of cost function.
"""
W = distance_to_weights(D)
C = np.cos(D)
def cost(Y):
"""Weighted Frobenius norm cost function."""
return 0.5 * np.linalg.norm(W * (C - np.abs(Y.T @ Y)))**2
def egrad(Y):
"""Derivative of the cost function."""
return 2 * Y @ (W**2 * (np.abs(Y.T @ Y) - C) * np.sign(Y.T @ Y))
return cost, egrad
def setup_CPn_cost(D, n, return_grad=False):
"""Cost using geodesic metric on CPn."""
W = distance_to_weights(D)
C = np.cos(D)**2
i_mtx = np.vstack((np.hstack((np.zeros(
(n, n)), -np.eye(n))), np.hstack((np.eye(n), np.zeros((n, n))))))
def cost(Y):
return 0.5 * np.linalg.norm(W * ((Y.T @ Y)**2 +
(Y.T @ (i_mtx @ Y))**2 - C))**2
if return_grad:
def egrad(Y):
J = np.vstack((np.hstack((np.zeros(
(n, n)), np.eye(n))), np.zeros((n, 2 * n))))
R = np.vstack((np.hstack((np.eye(n), np.zeros(
(n, n)))), np.zeros((n, 2 * n))))
return (Y @ (2 * (Y.T @ Y)) @ (2 *
(W *
((Y.T @ Y)**2 +
(Y.T @ (i_mtx @ Y))**2 - C))) +
(R @ Y + J @ Y) @ (2 * (Y.T @ J @ Y)) @ (2 * (W * (
(Y.T @ Y)**2 + (Y.T @ (i_mtx @ Y))**2 - C))))
else:
egrad = None
return cost, egrad
def setup_CPn_cost(D, n, return_grad=False):
"""Cost using geodesic metric on CPn."""
W = distance_to_weights(D)
i_mtx = np.vstack((np.hstack((np.zeros(
(n, n)), -np.eye(n))), np.hstack((np.eye(n), np.zeros((n, n))))))
def cost(Y):
return 0.5 * np.linalg.norm(
np.sqrt((Y.T @ Y)**2 + (Y.T @ (i_mtx @ Y))**2) - D)**2
return cost, None